I can definitely recommend the pawpaw. It's like pudding in a fruit. I've harvested from a remotely-located patch for years, but getting to that patch in late August/early September has been a challenge the last couple of years. I try to tell myself that it's okay to leave them for the bears, but no, screw the bears; they can eat grass or roots or something.
I think the global use for the other fruit is far more dominant, you guys should consider switching to using the term 'Asimina'. Wikipedia says some people call it 'prairie banana', too. I've eaten that somewhere but can't remember where. Not bad, but not as nice as https://en.wikipedia.org/wiki/Annona_squamosa which we get here also.
I haven't heard "paw-paw" in place of "papaya" outside of Asia, so I wouldn't call that global. We're not going to change the lyrics of all our old pawpaw-celebrating folk songs just to accommodate someone else's regional dialect. b^)